Observables in General Relativity Application to Cosmology Quantisation Conclusions & Outlook Manifestly Gauge – Invariant Relativistic Perturbation Theory Kristina Giesel Albert – Einstein – Institute ILQGS 25.03.2008 References: K.G., S. Hofmann, T. Thiemann, O.Winkler, arXiv:0711.0115, arXiv:0711.0117 K.G., T. Thiemann, arXiv:0711.0119 Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
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Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Manifestly Gauge – Invariant Relativistic
Perturbation Theory
Kristina Giesel
Albert – Einstein – InstituteILQGS
25.03.2008
References:K.G., S. Hofmann, T. Thiemann, O.Winkler, arXiv:0711.0115, arXiv:0711.0117
K.G., T. Thiemann, arXiv:0711.0119
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Plan of the Talk
Content
Application of Relational framework to General Relativity
Special Case of Deparametrisation: Two examples
Manifestly gauge-invariant framework for General Relativity
Application to Cosmology (FRW and perturbation around FRW)
Quantisation: Reduced Phase Space Quantisation
Conclusions & Outlook
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Problem of Time in General Relativity
Observables in General Relativity
Observables are by definition gauge invariant quantities
The gauge group of GR is Diff(M)
Canonical picture:
Constraints c,~c generate spatial and ’time’ gauge transformations
O gauge invariant ⇔ {c,O} = {~c,O} = 0
’Hamiltonian’ hcan for Einstein Equations is linear combination ofconstraints and thus constrained to vanish
Consequently: O gauge invariant ⇔ {hcan,O} = 0
Frozen picture, contradicts experiments, problem of time in GR
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Relational Formalism
Basic Idea [Bergmann ’60, Rovelli ’90]
Einstein Equations are no physical evolution equations
Rather describe flow of unphysical quantities under gauge transf.
Relational formalism:
Take two gauge variant f , g and choose T := g as a clock
Define gauge invariant extension of f denoted by Ff ,T in relation tovalues T takes
Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...
Solve αt(T ) = τ for t, then use solution tT (τ) for Ff ,T whichbecomes a function of τ
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Relational Formalism: Idea
f , g move along gauge orbit
PSfrag replacements
gauge orbit f gauge orbit g
f (t1)
f (t3)
g(t2)
g(t4)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Relational Formalism
Basic Idea [Bergmann ’60, Rovelli ’90]
Einstein Equations are no physical evolution equations
Rather describe flow of unphysical quantities under gauge transf.
Relational formalism:
Take two gauge variant f , g and choose T := g as a clock
Define gauge invariant extension of f denoted by Ff ,T in relation tovalues T takes
Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...
Solve αt(T ) = τ for t, then use solution tT (τ) for Ff ,T whichbecomes a function of τ
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Relational Formalism
Explicit Form for Ff ,T [Dittrich ’04]
Take as many clocks TI as they are CI then Ff ,T (τ) can beexpressed as powers series in T I with coefficients involving multiplePoisson brackets of CI and f .
Explicit form in general quite complicated
But: One has explicit strategy how to construct observables
Analysed in several examples, application to cosmology andcosmological perturbations [Dittrich, Dittrich & Tambornino]
Automorphism property
{Ff ,T (τ),Ff ′,T (τ)} = F{f ,f ′},T (τ),
If f (q, p) then Ff ,T = f (Fq,T ,Fp,T )
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Strategy of the Formalism
Steps to obtain EOM for observables
Consider a physical System for instance gravity & some standardmatter
We would like to derive EOM for the observables associated to(qa, p
a) of gravity & matter
Add additional action to the system which become clocks T
We havectot = cgeo + cmatter + cclock =: c + cclock = 0ctota = cgeo
a + cmattera + cclock
a =: ca + cclocka = 0
Construct observables wrt to these constraints: Fqa,T (τ) & Fpa,T (τ)
Construct so called physical Hamiltonian Hphys which generatestrue evolution of Fqa,T (τ), Fpa ,T (τ)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Special Case of Deparametrisation
Steps technically simplify
Deparametrisation: c tot and ctota can be solved for pclock
Expressions for Fqa/pa,T (τ) and Hphys simplify
Note: Hphys is in general different for each chosen clock system
Evolution of observables is generated by Hphys
EOM for observables are clock – dependent
Consider two examples for clarification:
scalar field without potentialk-essence
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Scalar field as a Clock (LQC-Model)
Deparametrisation for scalar field φ
Constraints:ctot = c(qa, p
a) + 12λ ( π2
√q
+ qabφ,aφ,b)
ctota = ca(qa, p
a) + πφ,a
Using ctota = 0 we get qabφ,aφ,b = 1/π2qabcacb (more details later)
Using c tot = 0 we get
π =√
| − √qλc −√
q√
λ2c2 − qabcacb| =: hφ(qa, pa)
Equivalent Hamiltonian constraint: c tot = π − hφ(qa, pa)
Construct observables Qa(τφ) := Fqa,φ(τ) and Pa(τφ) := Fpa,φ(τ)
Evolution:Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τφ) = {Hphys,P
a(τφ)}
Hφphys :=
∫
d3σ
√
| −√
QλC −√
Q√
λ2C 2 − QabCaCb|
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
K-essence (Thiemann ’06)
Deparametrisation for k-essence field ϕ: Case I
Constraints:ctot = c(qa, p
a) −√
[1 + qabϕ,aϕ,b][π2 + α2√q], α > 0ctota = ca(qa, p
a) + πϕ,a
Using ctota = 0 we get again qabϕ,aϕ,b = 1/π2qabcacb
Using c tot = 0 we get π = −hϕ(qa, pa)
hϕ(qa, pa) :=
√
12 (c2 − qabcacb − α2q) +
√
14 (c2 − qabcacb − α2q)2 − α2qabcacbq
Equivalent Hamiltonian constraint: c tot = π + hϕ(qa, pa)
Construct observables Qa(τϕ) := Fqa,ϕ(τ) and Pa(τϕ) := Fpa,ϕ(τ)
Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τϕ) = {Hphys,Pa(τφ)}
Hϕphys :=
∫
d3σhϕ(Qa,Pa)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
K-essence (Thiemann ’06)
Deparametrisation for k-essence field ϕ: Case II
Constraints:ctot = c ′(qa, p
a) −√
[1 + qabϕ,aϕ,b][π2 + α2√q], α > 0ctota = c ′
a(qa, pa) + πϕ,a
Using ctota = 0 we get again qabϕ,aϕ,b = 1/π2qabc ′
ac′b
Using c tot = 0 we get π = −h′ϕ(qa, p
a)h′(qa, p
a) :=√
12 ((c ′)2 − qabc ′
ac′b − α2q) +
√
14 ((c ′)2 − qabc ′
ac′b − α2q)2 − α2qabc ′
ac′bq
Equivalent Hamiltonian constraint: c tot = π + h′ϕ(qa, p
a)
Construct observables Qa(τϕ) := Fqa,ϕ(τ) and Pa(τϕ) := Fpa,ϕ(τ)
Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τϕ) = {Hphys,Pa(τφ)}
Hϕphys :=
∫
d3σh′ϕ(Qa,P
a)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup